LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 16 3: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0
4 23 4: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15 using the following ranking functions, which are bounded by −19.

8: 0
6: 0
7: 0
2: 0
3: 0
4: 0
5: 0
0: 0
1: 0
8: −7
6: −8
7: −9
2: −10
3: −10
4: −10
5: −10
3_var_snapshot: −10
3*: −10
4_var_snapshot: −10
4*: −10
0: −13
1: −14

3 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3* 19 3: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0

4 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3 17 3_var_snapshot: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0

5 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

4* 26 4: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0

6 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

4 24 4_var_snapshot: x2_post + x2_post ≤ 0x2_postx2_post ≤ 0x2_0 + x2_0 ≤ 0x2_0x2_0 ≤ 0x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX5_post + oldX5_post ≤ 0oldX5_postoldX5_post ≤ 0oldX5_0 + oldX5_0 ≤ 0oldX5_0oldX5_0 ≤ 0oldX4_post + oldX4_post ≤ 0oldX4_postoldX4_post ≤ 0oldX4_0 + oldX4_0 ≤ 0oldX4_0oldX4_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

7.1 SCC Subproblem 1/1

Here we consider the SCC { 2, 3, 4, 5, 3_var_snapshot, 3*, 4_var_snapshot, 4* }.

7.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 10.

2: 3 + 7⋅x2_0
3: −2 + 7⋅x1_0
4: 1 + 7⋅x0_0
5: −1 + 7⋅x0_0
3_var_snapshot: −3 + 7⋅x1_0
3*: −2 + 7⋅x1_0
4_var_snapshot: 7⋅x0_0
4*: 2 + 7⋅x0_0

7.1.2 Transition Removal

We remove transitions 17, 19, 24, 26, 1, 2, 5, 6 using the following ranking functions, which are bounded by −4.

2: 3
3: −3
4: 1
5: −1
3_var_snapshot: −4
3*: −2
4_var_snapshot: 0
4*: 2

7.1.3 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.3.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 16.

7.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

7.1.3.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 23.

7.1.3.2.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

Tool configuration

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